PHYSICAL REVIEW E 94, 063002 (2016)
Inverted cones and their elastic creases
Keith A. Seffen*
Advanced Structures Group Laboratory, Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
(Received 30 August 2016; revised manuscript received 21 October 2016; published 19 December 2016)
We study the elastic inversion of a right circular cone, in particular, the uniform shape of the narrow crease
that divides its upright and inverted parts. Our methodology considers a cylindrical shell analogy for simplicity
where the crease is the boundary layer deformation. Solution of its governing equation of deformation requires
careful crafting of the underlying assumptions and boundary conditions in order to reveal an expression for the
crease shape in closed form. We can then define the characteristic width of crease exactly, which is compared
to a geometrically nonlinear, large displacement finite element analysis. This width is shown to be accurately
predicted for shallow and steep cones, which imparts confidence to our original assumptions. Using the shape of
crease, we compute the strain energy stored in the inverted cone, in order to derive an expression for the applied
force of inversion by a simple energy method. Again, our predictions match finite element data very well. This
study may complement other studies of creases traditionally formed in a less controlled manner, for example,
during crumpling of lightweight sheets.
DOI: 10.1103/PhysRevE.94.063002
I. INTRODUCTION
Indenting a thin, doubly curved elastic surface can result
in local inversion, where a dimple forms and spreads over the
surface. For the case of a point-loaded spherical cap, the dimple
is axisymmetrical and bounded by a planar circular ridge. In
the limit of zero cap thickness, the ridge acquires infinitesimal
width as the bending stiffness diminishes; furthermore, the
dimple is a perfect reflection of the initial region above the
current ridge plane. This simplified but immediate view of
the partially inverted shape was first proposed by Pogorelov
[1]. As we increase the thickness, the ridge circle expands to
become a smooth open torus, or crease, which seamlessly interconnects the upright and inverted parts. Tracing out a meridian
on the surface gives a clear indication of the crease extent, but
where it precisely begins or ends is not immediately obvious.
Most of the elastic deformation, however, is concentrated in
the crease, so knowing its shape precisely is key to finding
other features of the problem such as the applied forces in
equilibrium. Often, the crease is endowed with a uniform shape
because of its much smaller size, usually a constant radius of
curvature [2], which allows the crease width to be simply
written in terms of its meridional arc length or its projected
width onto the ridge plane. A straightforward dimensional or
energy analysis is then possible with the familiar scaling law
that width depends on the square root of the product of shell
thickness and spherical radius. Determining the constant of
proportionality in this expression typically requires data from
experiments or numerical simulations, so the absolute width
is often an approximate, if dimensionally correct, expression.
A more recent and thorough examination of the Pogorelov
proposition appears in [3], which also furnishes a historical
review of spherical cap inversion. They revise upward the number of latitudinal regimes to seven, with five alone describing
the inverted shape between the central loading point and the
ridge, followed by the ridge itself and the original part beneath.
These regimes are found from an elegant asymptotic analysis
*
[email protected]
2470-0045/2016/94(6)/063002(9)
of the governing equation of axisymmetrical deformation,
which is compared against its full numerical solution and finite
element analysis. There is a richness of results; however, the
crease extent is expressed only in order-of-magnitude terms
because it is not of explicit interest. Here we examine the
precise shape of a similar crease formed during inversion of a
right circular cone. Figure 1 indicates some partially inverted
shapes obtained from the finite element analysis of the next
section. Three major geometrical regimes are evident: upright
and inverted sections, which are largely undeformed, and a
narrow crease. We can surmise a Pogorelov-type profile in
the limit of zero thickness, where the cutaway in Fig. 1(c)
shows the more heavily strained area pertaining to the crease
diverging from this profile superimposed. Axisymmetry is
prescribed at all times during computation even though new
studies suggest the possibility of secondary buckling of the
circular ridge into a rough polygonal outline in practice [4].
We will be concerned only with solving for the shape of an
axisymmetrical crease using the linear governing equation of
deformation for a cylindrical shell.
Such an approach would appear to flout the major effects
of geometrical nonlinearity and initial cone shape upon the
crease shape, but it greatly reduces the complexity of analysis
and enables accurate closed-form solutions provided two key
assumptions are upheld. First, a Pogorelov viewpoint for a
partially inverted cone of zero thickness is applicable; this
then provides a coordinate reference for actual displacements.
Second, the meridional span of major displacements around
the ridge line is very short such that the curvature of
the underlying Pogorelov reference remains approximately
constant, i.e., cylindrical, over this span. The validity of this
approach is then tested in two subtle ways. When the cone is
steep, it is naturally close to a cylinder, but across the ridge
there can be large rotations, up to 180◦ , depending on the
initial geometry; on the other hand, a shallow cone is less
like a cylinder but has much smaller displacement gradients
in keeping with the shallow shell requirement of the linear
governing equation. Remarkably accurate results, however,
emerge for a range of conical geometries. We then determine
an expression for the inverting force applied to the apex, which
063002-1
©2016 American Physical Society
KEITH A. SEFFEN
PHYSICAL REVIEW E 94, 063002 (2016)
(a)
(d)
(c)
(b)
FIG. 1. Elastic inversion of a thin conical shell using finite element analysis. The initial cone angle, measured from the vertical, is 60◦
and the side length of cone is 50 mm; the apex is a small spherical cap of radius 3 mm and everywhere the thickness is 0.1 mm. (a) Initial
configuration where a force is applied to the apex and the base of the cone is held rigid. (b) Two views of the partially inverted state, highlighting
contours of maximum principal strain. These are focused in the original apex and around the ridge that separates the inverted part from the
original upright part. (c) Cutaway view of (b) from afar and close up, showing the deviation of the conical shape near the ridge line away from
perfectly straight meridians (gray lines). The width of this deviation is roughly the span over which the strain contours diminish in variation
and is shown by the arrow: This is the conical crease associated with the ridge line. (d) Almost fully inverted cone, indicating the extent of
deformation during analysis.
is usually intractable in other elastic studies: During plastic
inversion of metal cones and other axisymmetrical shells,
where crease localization can occur too, the driving force is
usually amenable in closed form, but the underlying formation
and propagation of creases are very different from here (see,
e.g., [5]).
We will not quantify the formation stage for this itself
is challenging. For example, conical inversion begins with
turning the sharp apex inside out, which is impractical and
leads to computational difficulties because of the singular
geometry. However, we must marshal inversion in some way,
so we cheat at the very start by smoothing the apex to form a
very small, localized spherical cap [6]. Under a central point
force, this region locally inverts early on in the deformation
profile, enabling conical inversion to become well developed,
the starting point of our study. We consider a range of cone
angles, including a tube in the extreme, and we highlight
representative behavior, in particular, our definition of crease
width according to how the ridge strains attenuate. We then
present our analytical model and compare predictions with
finite element data before concluding with a brief discussion.
II. FINITE ELEMENT ANALYSIS
The commercial software package used in Fig. 1 and
throughout is ABAQUS [7]. Elements are axisymmetrical twonode linear SAX1 elements available from the standard library
and each conical model has 500 elements along the defining
meridian; trials using fewer three-node quadratic elements
makes little difference to the computational efficiency. The
material has a typical Poisson ratio of 0.3 and a Young modulus
of 1 MPa, which is relatively soft but intends to mimic a soft
rubber, and all thicknesses are of the order of 0.1 mm for a cone
side length of around 50 mm. The cone apex is replaced by a
spherical cap of radius 3 mm, which connects smoothly to the
rest of the cone; a vertical force is applied normal to the cap at
its pole and the base of the cone is fully built in. Geometrical
nonlinearity is coupled with a Riks arc length algorithm [7]
during solution to capture fine, sometimes highly nonlinear
features of the load path. Before examining a typical path,
consider the four partially inverted cones in Fig. 2. They begin
with a shallow cone in Fig. 2(a) inclined initially to the vertical
by 60◦ and become increasingly steeper with cone angles of
45◦ , 30◦ , and 0◦ , which is a cylindrical tube. Each panel also
reveals the deformed cross section where the crease profile is
clearly evident and accentuated by highlighting elastic strain
contours; each inverted cap is also highly strained. A sense
of mirror symmetry about the ridge plane is also conveyed,
in particular for the tube, where the inverted part is passing
through its undeformed self. We have also added the Pogorelov
outline to each panel, in order to highlight the crease shape and
its relative size as the cone angle decreases.
Representative results from the first case are given in Fig. 3.
First, the inverting force F is plotted against the displacement
of the apex d in Fig. 3(a). Although difficult to see, the response
is linear at first before softening as the apical cap inverts fully,
with the crease forming completely soon after. The crease then
rolls down the side of the cone as the applied force decreases
exponentially. When the crease is halted by the built-in base,
the cone is almost inverted; at this stage, the crease mainly
stretches in plane to accommodate d increasing, leading to a
rapid rise again in F albeit with a small intermittent dip.
At some intermediate displacement, we plot the hoopwise
strain h in Fig. 3(b) as a function of the intrinsic coordinate
s from the apex [Fig. 4(c)], which is the same for every
meridian. Moving out of the inverted cap, there are high
but localized strains that diminish quickly before increasing
again midmeridian at the position of the crease. There is first
compressive hoop strain on the inside of the crease facing
the inverted apex, followed by tensile strain on the outside.
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INVERTED CONES AND THEIR ELASTIC CREASES
PHYSICAL REVIEW E 94, 063002 (2016)
(a)
(b)
(c)
(d)
FIG. 2. Partial inversion of four different cones using finite elements. The top row shows full cones partially inverted. The middle row
shows cutaway counterparts highlighting the crease width around the ridge line according to color changes in the maximum principal strain
contours. The bottom row shows axisymmetrical meridians (green) compared to the perfectly straight meridians (gray), showing how the actual
crease region diverges and detaches from the latter; the crease width is later defined to be the arc length of the detached region. The cone angles
are (a) 60◦ , (b) 45◦ , (c) 30◦ , and (d) 0◦ ; other material and geometric properties are the same as Fig. 1. In (d) the inverted part overlays the
original undeformed tube, giving a mottled complexion of interference.
Such compression is surmised as a mechanism for secondary
buckling of the crease, but we note a strongly antisymmetrical
profile about a local origin that exactly coincides with the ridge
line. The width of the crease can be clearly correlated with the
span of major strain activity, and we choose bounding points
to be given by h first, becoming zero again on either side of
−3
1.5
x 10
(a)
−3
x 10
the ridge line. The corresponding meridional span is denoted
by 2l ∗ , which will be compared to the same definition of width
from theory. Notice also that the highest strain is around 0.5%,
which exceeds the yield strain of most metals; it is even higher
at the start of inversion but decreases as inversion proceeds.
Longitudinal strains are smaller but always vary in proportion
(b)
−3
8
2 l */smax
l* [m]
0
0.5
0
0.02 0.04
d [m]
0
4
2
−5 cap
0
(c)
6
1
εh
F [N]
5
x 10
0.5
s / smax
1
0
d [m]
0.05
FIG. 3. Properties of the conical inversion of Fig. 1. (a) Inverting force F vs apex displacement d. Three regimes are highlighted: first,
initial inversion up to d ≈ 0.006 m, where the apical cap turns inside out and the crease on the ridge line becomes fully formed; second,
movement of the crease through the undeformed cone as inversion proceeds up to d ≈ 0.04 m; third, cessation of inversion when the crease
reaches the base of the original cone. (b) Hoopwise strain h along a meridian at some point during inversion when the crease is well formed.
The meridional coordinate from the apex is s [see Fig. 4(c)] and smax corresponds to the base of the cone. Up to s/smax ≈ 0.1, the strains belong
to the original apical cap; around s/smax = 0.5, we encounter strains in the crease, which diminish quickly on either side. This pronounced
variation gives a strong indication of the extent of the crease: It is antisymmetrical about the ridge line with h = 0 on the ridge line itself; after
peaking and reaching zero on either side, h quickly diminishes. We define the width of the crease by the span 2l ∗ between these zero points
around the ridge line. Green circles are distinct nodal positions along the meridian from finite element analysis, which are used to calculate this
width at each increment of solution. (c) Variation of l ∗ with d during inversion. The initial data before d ≈ 0.005 m are not shown because the
crease has not yet separated from the apical cap. Note the diminishing rate by which l ∗ increases.
063002-3
KEITH A. SEFFEN
PHYSICAL REVIEW E 94, 063002 (2016)
α
Q
T
γ
b
d
R
r2
l*
r2
α α
γ
M
w
r
γ
w
s
x
x
CL
(a)
CL
(b)
(c)
A
(d)
FIG. 4. Schematic rendering of a partially inverted cone and its ridge line crease. (a) Pogorelov viewpoint of deformation in which the cone
inverts symmetrically under an axial force. The small circles signify the position of the inverting ridge line. (b) Meridional geometry for perfect
inversion. The heavy black line is a cone with a sharp apex; the gray line is our cone with a shallow apical cap whose vertical displacement is
d. For both, the position of the ridge line is a small circle located a distance b from the edge of the cap and sitting at radius r from the centerline
(CL) axis of revolution. The radius of hoopwise curvature at the ridge for a cone is r2 , which extends from the centerline to the undeformed
meridian immediately below the ridge; γ is a dummy angle equal to π/2 − α, where α is the cone angle. (c) Shown on top is the definition of
cap geometry from (b), where R is its radius of curvature; s is an intrinsic coordinate from the top of the cap. The bottom shows a close-up
view of the crease geometry around the ridge line. The underlying straight axis is the Pogorelov shape from (a), which intersects the crease
ridge line. The local coordinate x is measured from the ridge line on one side only; x = l ∗ is the characteristic half length of the crease when
the transverse displacement w becomes zero again. (d) Half of the crease from (c) with free-body forces and bending moment added to the
position of the cut at x = 0. The positive directions of M and Q give dM/dx = Q.
to hoop strains by the Poisson ratio, which suggests negligible
meridional stresses compared to hoop stresses.
For each solution increment, we detect positions of zero
strain in order to find the variation of l ∗ with d, which is plotted
in Fig. 3(c). We omit the data at small displacements because
the crease has not yet localized, giving a false impression
of its size. After becoming well formed, l ∗ behaves in a
regular way, increasing with d at a diminishing rate. Such
variation is repeated for many conical shapes and thickness,
where the radius-to-thickness ratio of shell can range from
15 at the apex to nearly 1300 close to the base. We aim to
predict the responsible factors for l ∗ and so must carefully
focus on the kinematic features in our model. However, we
also wish to find the force required to pull the crease through
the cone. For example, it is not clear why the inverting force
should decrease: At the same time, the crease circumference
increases, suggesting that more of the cone is becoming
strained, viz., more external work and a higher force. We
resolve this surmised discrepancy in the following analysis.
III. THEORETICAL MODEL
Figure 4(a) shows a cross section of the equivalent
Pogorelov viewpoint of conical inversion. The inverted part
can be reflected in the ridge plane to yield the original
undeformed cone and all lines are straight. This is the zero
strain configuration because there are no changes in length
anywhere and zero shell thickness precludes any stored elastic
energy in the ridge line. More geometrical detail within the
meridional plane is furnished in Fig. 4(b) for a cone with a
sharp apex as well as our apical cap. This cone is also perfectly
inverted under a polar displacement d with the crease, still of
zero width, being located at a distance b along the initial
meridian from the end of the cap. Using Figs. 4(b) and 4(c),
where the cone angle is denoted by α and the cap radius by R,
simple geometry affords
d = 2R(1 − sin α) + 2b cos α.
(1)
The current latitudinal radius at the crease position is r, which
is equal to R cos α + b sin α. At the same position, the current
radius of hoopwise curvature of the conical shell is measured
by the distance from the vertical conical axis normal to the
meridian, where the usual notation is r2 from [8]. If we select
the meridian in the upright part, then r2 is drawn as per Fig. 4(b)
and r/r2 = cos α. An explicit expression for r2 in terms of d
can now be found by eliminating b between them:
tan α d
− R(1 − sin α) .
(2)
r2 = R +
cos α 2
This returns r2 = d tan α/2 cos α for a perfect cone when R
is zero. Strictly speaking, the radius of curvature of the actual
ridge line is not defined because of the discontinuity in gradient
there and the expression for r2 is only true immediately on
either side of the ridge (if we had selected the inverted meridian
instead, then r2 changes direction and lies above r). However,
this presents no problem because at the ridge line itself, we
need to interrogate the local crease shape in order to ascribe
curvature properties in detail.
Figure 4(c) shows a schematic axisymmetrical cross section
of a crease, which is drawn relative to the Pogorelov outline.
Displacements away from this outline constitute straining and
curving, where the normal component w is assumed to be
dominant. We also assume that w is symmetrical about the
ridge line because the crease is far enough away from the
base or apex. The formulation can now be contracted slightly
by defining a positive meridional coordinate x moving away
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INVERTED CONES AND THEIR ELASTIC CREASES
PHYSICAL REVIEW E 94, 063002 (2016)
from the ridge towards the base; x does not take negative
values, rather, any functions of it are merely reflected about
x = 0 either symmetrically or antisymmetrically provided we
are careful about boundary conditions at x = 0. The hoopwise
strain h is therefore w/r2 (x) from [8], where r2 (x) is the
variation in the underlying conical radius of curvature beyond
the ridge line, found by substituting b with b + x into Eq. (1)
and so forth. The reason for the crease touching the Pogorelov
ridge line at x = 0 is now apparent, for here w must be zero,
which sets the hoop strain to be zero as per Fig. 3(b). Beyond
the ridge line, r2 (x) increases, but if w decreases sharply with
x then there is little change in r2 within the crease itself: In
other words, the radius of hoopwise curvature of the crease
is everywhere 1/r2 , with r2 expressed by the current value of
Eq. (2). As noted, meridional strains are not negligible, but
w does not depend on them directly. The curvature, however,
in this direction κ is set by the shallow gradient expression
κ = −d 2 w/dx 2 , where absolute differentials assert the onedimensional nature of problem.
We idealize further by excising half of the crease in Fig. 4(c)
for all of x and redrawing it in Fig. 4(d). At the cut x = 0 and
thus everywhere on the ridge line, there must be a bending
moment M and locally aligned forces in shear Q and in tension
T ; these are all axisymmetrical quantities defined per unit
length of shell and are drawn in their positive directions. Even
though we do not show the far field forces at the base in this
view, it is possible to write certain equilibrium relationships
between them and M, Q, and T . However, specified in this
way, our problem turns out to be statically indeterminate
and we must invoke arguments of geometrical compatibility
for a complete solution. Very quickly, the level of algebraic
complexity increases, which is not the aim of study. Instead,
we assume that T is not directly responsible for transverse
displacements, so we neglect its effect for now, and that M
and Q are wholly responsible for ensuring the local boundary
conditions at x = 0. We may therefore focus on the interaction
of M and Q alone and, because r2 is deemed to be constant
everywhere along the crease, they behave as if they are loads
applied to the edge of a cylinder of uniform radius r2 .
In the absence of a normal pressure, the governing equation
of transverse deflections for a cylindrical shell is given by [8]
d 4w
Et
+
w = 0.
4
dx
Dr22
(3)
This is derived by considering equilibrium of an axisymmetrical element of original hoopwise curvature 1/r2 . Linear
elasticity is specified and the deformation is controlled by
meridional bending where M = Dκ and by hoopwise straining
via h : D is the standard flexural rigidity equal to Et 3 /12(1 −
ν 2 ), where E is the Young modulus, ν is the Poisson ratio, and
t is the transverse shell thickness. A slightly more compact
notation replaces the w prefactor with β 4 = Et/4Dr22 and
we try solutions of the form w = Aeξ x , where the unknown
amplitude A is found later from the boundary conditions.
The corresponding auxiliary√ equation sets ξ 4 + 4β 4 = 0,
returning roots of ξ = (±1 ± −1)β and a general solution of
four terms as A1 eβx sin βx + A2 e−βx sin βx . . . . Those terms
involving exponential growth can be discounted for w rapidly
decaying with x, resulting in two terms whose amplitudes are
found by setting M = Dκ and Q = dM/dx at x = 0:
Q
e−βx
M(sin
βx
−
cos
βx)
+
cos
βx
for x 0.
w=
2β 2 D
β
(4)
Ensuring that w = 0 at x = 0 obviates M = Q/β, giving a
gradient of M/2βD at the same position, which must equal
π/2 − α from Fig. 4(d). The final expression for w turns out
to be rather compact:
w=
(π/2 − α) −βx
sin βx
e
β
for x 0.
(5)
Because of the proportionality between w and hoop strain h
and given the latter’s variation in Fig. 3(b) on either side of
the ridge line, we could have inferred the expression above
by inspection. The formal analysis is conclusive and Eq. (5)
immediately offers l ∗ when we set h , and hence w, equal to
zero at x = l ∗ , giving sin βl ∗ = 0 and βl ∗ = π . Recalling our
previous definitions of β and D, we can therefore write
1/4
1/2
4Dr22
π r2 t 1/2
l∗ = π
⇒ l∗ =
Et
[3(1 − ν 2 )]1/4
√
(l ∗ ≈ 2.44 r2 t for ν = 0.3).
(6)
This confirms that the crease increases in width during
inversion because r2 increases with d according to Eq. (2).
Comfortingly, the square-root dependence of l ∗ upon r2 points
to the cusplike variation we expect, in particular, when R is
small enough to be ignored, r2 is proportional
to d for a given
√
cone angle, and thus l ∗ varies with d. In general, being
proportional to the square root of the product of thickness and
local radius of curvature, l ∗ is akin to the width of boundary
layer deformation along the edges of long, curved thin-walled
strips [9] and to the width of periodic compression wrinkles in a
variety of pretensioned membranes [10,11], as discussed later.
We compare Eq. (6) to finite element data in Fig. 5 first for
a fairly shallow cone (α = 60◦ ) of three different thicknesses,
t = 0.2, 0.1, and 0.067 mm. For the apical cap, R = 3 mm and
the straight meridian is 50 mm long. We plot dimensionless
quantities l ∗ /t vs d/dmax , where d is Eq. (1) and d = dmax
when the cone is fully inverted. Again, early computational
data near d = 0 are omitted because the crease has not yet
properly formed. Otherwise, the predictions by Eq. (6) are
almost indistinguishable from those of finite elements. We also
plot the variation of l ∗ /r2 against d/dmax to determine how the
crease width compares to the local radius of conical curvature.
The ratio l ∗ /r2 is always a maximum initially when r2 is smallest and is typically around 0.25–0.4, giving a crease length 2l ∗ ,
just slightly less than r2 . However, the ratio quickly attenuates
to around 0.1–0.15, which implies that r2 changes little over
the crease span, thereby confirming our original proposal.
Figure 6 deals with different cone angles of the same
thickness and, again, the differences between theory and
computational data are minimal. Even though a steeper cone
implies a slower rate of change of r2 along a meridian, which
bolsters the cylindrical shell analogy, the initial gradient in
Fig. 4(d) at x = 0 also increases, which undermines the shallow shell assumption. For a tube where α = 0◦ , this gradient is
infinite yet Eq. (6) differs by less than 10% compared to finite
063002-5
KEITH A. SEFFEN
PHYSICAL REVIEW E 94, 063002 (2016)
(a)
(b)
(c)
50
80
60
l*/t
60
30
l*/t
l*/t
40
40
20
20
10
0
0
20
0
0
0.5
d/d max
0.8
40
0
0
0.5
d/d max
0.5
d/d max
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0.2
0
0
0.4
l*/r2
l*/r
2
l*/r2
0.6
0.1
0.5
d/d
0
0
0
0
0.5
d/d
max
max
0.5
d/d
max
FIG. 5. Comparison of l ∗ from Eq. (6) (blue circles) with finite element data (solid red line). Each vertical pair of panels is for the same
indicated thickness (a) t = 0.2 mm, (b) t = 0.1 mm, and (c) t = 0.067 mm and all have the same cone angle α = 60◦ and the same apical cap
radius R = 3 mm; the Young modulus is 1 MPa and the Poisson ratio is 0.3. The top row compares the crease width l ∗ to shell thickness t; the
bottom row considers l ∗ with the local radius of conical curvature r2 . The apex displacement is d, which reaches a maximum value dmax when
the cone is fully inverted. The early parts of all finite element data do not produce distinct trends because the crease is not properly formed and
so are omitted for clarity.
element data, which is remarkable. In this case, r2 is a constant
and equal to the radius of the original apical cap. Note that in
general, as the cone angle decreases, so does l ∗ .
We now proceed to finding the inverting force F using
an energy formulation for the entire cone based on the strain
energy stored in the crease and not in the deformed apical cap.
This process is reasonably light on algebra because w is a fairly
simple expression and there are only two major strain energy
components; as we remarked before on statical indeterminacy,
finding F by equilibrium, etc., becomes unwieldy very quickly.
First, let the amplitude of w in Eq. (5) be A for convenience,
which gives
h =
w
A
d 2w
= e−βx sin βx, κ = − 2 = 2Aβ 2 e−βx cos βx.
r2
r2
dx
(7)
An expression for the strain energy density per unit surface area
of cone may now be distilled from the general expression for
shells found in [8] by assuming that these kinematic terms are
predominant and remembering that in-plane meridional strains
are equal to −νh . Hoopwise curvature changes are negligible
compared to κ and, because of axisymmetry, shear stresses
and twisting curvatures are zero. The simplified density can be
verified as (1/2)Eth2 + (1/2)Dκ 2 and the total strain energy
is then found by integrating this expression over the conical
surface area where, recall, x is only positive. The integration
can be simply doubled because the crease shape is perfectly
symmetrical about the ridge line. Its limits are clearly zero
and x = l ∗ , but if we set the upper limit to infinity instead, the
resulting expressions are greatly simplified but remain accurate
because of the rapid attenuation of w. To this end, we calculate
the identities
x=∞
x=∞
1
3
,
(e−βx sin βx)2 =
(e−βx cos βx)2 =
8β
8β
x=0
x=0
(8)
and note that a hoopwise elemental strip has an area of
2π r2 cos αdx where, as per usual, we set r2 equal to its value
at the ridge line. Altogether, we find the total strain energy
stored in the cone U to be
063002-6
2π EtA2 cos α
+ 3π r2 DA2 β 3 cos α
8r2 β
for A = (π/2 − α)/β.
U =
(9)
INVERTED CONES AND THEIR ELASTIC CREASES
(a)
PHYSICAL REVIEW E 94, 063002 (2016)
(b)
20
20
0.2
0.4
d/d
0.6
0
0
0.8
0.02
0
0.2
(c)
0.4
d/d
0.6
0.8
0.04
0.02
0.01
max
0
100
200
d/t
max
300
0
400
400
F/Et
10
2
0.1
F / E t2
l*/t
*
d/t
(d)
15
20
200
0.1
40
30
0
(c)
(d)
50
F/Et
F / E t2
l*/t
*
l /t
40
2
0.03
40
l /t
(b)
0.06
60
0
0
(a)
0.04
60
0.05
0.05
5
10
0
0.2
0.3
d/d max
0.4
FIG. 6. Comparison of l ∗ from Eq. (6) (blue circles) with finite
element data (solid red line) for different indicated cone angles:
(a) α = 60◦ , (b) α = 45◦ , (c) α = 30◦ , and (d) α = 0◦ . The shell
thickness for all is 0.1 mm, the apical cap has a radius of 3 mm, and
the material properties are the same as in Fig. 5. Again, early finite
element data do not apply and are left out, and because (d) is a tube,
we do not show all of l ∗ with respect to d because it is practically
constant.
The total potential energy for the system V is defined as the
difference between the internal strain energy and quantities of
external work. Since the point of application of F moves by d
in the same direction, the latter is simply F d and V = U − F d.
We may retain d in its general form, but for compactness we
set R to be zero, giving d = 2r2 cos α/ tan α. The expression
for V is now
1/2
π cos α(π/2 − α)2 Et 5/2 r2
V =
33/4 (1 − ν 2 )3/4
2F r2 cos α
−
tan α
200
d/t
400
0
600
0
200
400
d/t
600
FIG. 7. Comparison of the inverting force F from Eq. (11) (blue
circles) with finite element data (solid red line) under displacement d
for the indicated cone angles α: (a) α = 60◦ , (b) α = 45◦ , (c) α = 30◦ ,
and (d) α = 15◦ . The Young modulus E is 1 MPa and the Poisson
ratio ν is 0.3; all thicknesses are t = 0.1 mm.
d/t to compare directly with Eq. (11), although it can easily
be d/dmax , as in the other figures. Because of the reciprocal
term d −1/2 , F must decay even though we initially thought
it may increase as the crease envelopes more material during
displacement. Note that Fig. 7 deals with the same thickness,
0.1 mm; analysis for more thicknesses show the same close
trends but are not included for brevity’s sake. We replot the
data from Fig. 7 in Fig. 8 using logarithmic axes to ascertain
the power-law variation of the computational data assuming
(b)
(a)
(10)
after substituting for A and replacing β with π/ l ∗ . Now F is
found by minimizing V with respect to the generalized coordinate r2 and setting equal to zero. After some manipulation and
reexpressing r2 in terms of d, we can write the final expression
for F in one dimensionless form as
F
π (sin α)1/2 (π/2 − α)2 t 1/2
=
.
2
Et
23/2 33/4 (1 − ν 2 )3/4 d 1/2
0
10
−2
−2
10
1
10
(11)
We compare this to finite element data in Fig. 7 for cones
of increasing steepness up to α = 15◦ . We cannot compare
meaningfully to the case of a tube (α = 0) because Eq. (11)
returns zero force owing to the sin α term when the finite
element analysis gives a nonzero but constant force. The reason
for the latter stems from having an apical cap, where a force
must be applied to keep it inverted even though there is no
energy penalty as the crease moves along the tube, which
is reflected by Eq. (11). For the cone angles in Fig. 7, we
include the computational data during the formation stage, in
order to highlight the much higher forces at this time. When
the crease becomes well formed afterward, the correlation is
delightfully close despite the data beginning to diverge for
the steepest cone. We plot the dimensionless displacement as
F / E t2
0.1
10
−1
2
d/t
2
10
10
d/t
(c)
(d)
−1
10
F / E t2
0
0
2
0.8
F/Et
0.4
0.6
d/d max
2
0.2
F/Et
0
0
2
10
d/t
10
2
d/t
FIG. 8. Same force-displacement data as in Fig. 7 plotted on
logarithmic axes: (a) α = 60◦ , (b) α = 45◦ , (c) α = 30◦ , and (d)
α = 15◦ . As before, blue circles are for theory and solid red lines for
finite element data. Solid green lines pertain to the best power-law fit
of the finite element data for a well-formed crease where it is assumed
that F ∝ d m ; the corresponding index m, which is also the gradient of
green lines, is (a) −0.566, (b) −0.543, (c) −0.521, and (d) −0.506.
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KEITH A. SEFFEN
PHYSICAL REVIEW E 94, 063002 (2016)
that such a variation is presently applicable. We process only
the data beyond formation of the crease and we then add the
best fit predictions to the figure; for each case, the power-law
index, tantamount to the gradient of the data, is given. There
are two comforting findings. First, the force vs displacement
data are linear in the logarithmic sense, thereby confirming the
sensibility of a power-law variation. Second, all indices are
very close to −0.5, as suggested by Eq. (11), which reinforces
the credibility of this formula and underlying assumptions.
IV. DISCUSSION
Despite employing a linear governing equation of deformation for shallow shells, our analysis accurately captures the
steady state crease size and inverting force over a range of
initial conical geometries. It is especially pertinent for small
conical angles where, despite the initial geometry being close
to cylindrical, the boundary condition for rotation is evidently
not small. We note first that l ∗ does not depend on this rotation.
Second, the form of the shallow transverse deflections w
in Eq. (5) turns out to be a rather good fit for the large
rotation nature of this specific problem for several reasons.
From the arctangent of the gradient at x = 0 we may compare
the corresponding boundary rotation against the actual value
π/2 − α. For a pure cylinder (α = 0◦ ), dw/dx = π/2, giving
a rotation of arctan(π/2) ≈ 58◦ ; however, as the cone angle
increases, the discrepancy between these angular measures
quickly reduces, e.g., when α = 45◦ , we see a rotation of
arctan(π/2 − π/4) = 38◦ . Moderate rotations are therefore
described very well. Away from the boundary, the proportions
of crease afforded by w also compare well to finite elements:
For example, the ratio of wmax to l ∗ is largest for a cylindrical
tube and is roughly equal to one-fifth from finite element data;
Eq. (5) predicts 0.5 exp (−π/4) sin π/4 ≈ 1/6. The inverting
force (11) stems from a Rayleigh-Ritz-type energy analysis,
which is known to be accurate if the selected mode shape is
close to actual deflections, especially so when w comprises
a single term. Of course, we could approach the problem
within the spirit of large deflections from the outset. The
corresponding governing equation is a cylindrical version of
the well-known beam elastica, where geometrical nonlinearity,
particularly in view of axial force effects, is central to its
expression. However, for our problem, the effect of axial forces
is not dominant because their stresses are much smaller than
circumferential stresses; this can be seen in the shape of the
crease about the ridge line, which is virtually symmetrical
because in-plane meridional stresses are minimal (but not
zero). As noted initially, many scaling analysis do not consider
the detailed variation of crease shape, which does not reveal
the prefactor for any characteristic length, even though its
dimensional dependences are clear: Suffice to say, we have
established the prefactor immediately.
[1] A. V. Pogorelov, Bending of Surfaces and Stability of Shells,
Translations of Mathematical Monographs (American Mathematical Society, Providence, 1988).
Our linear analysis is also helped by the crease being a
relatively small but concentrative feature within the overall
geometry, which allows us to simplify its local performance; it
is also a contrived example that permits controlled formation
of a uniform crease. Apart from the case of spherical inversion
with its own circular crease, similar problems in the literature
usually focus on creases formed in an unregulated way, e.g.,
when a paper sheet is manually crumpled into a ball. If
the ball is unraveled into a planar sheet again, the crease
distribution is naturally random but all creases are straight [12].
As such, we may be tempted to think that these are physically
different creases compared to ours; however, their analysis
often embraces the same competition between local bending
and stretching where the simplest stress-free crease begins as
a narrow cylindrical strip before being bent into a shallow
saddle along its axis in order to mimic the effect of global
bending [13]. The strip becomes doubly curved, which incurs
stretching according to Gauss’s theorema egregium [8], and the
inevitable balance between these deformation modes results
in a natural radius of spanwise curvature for the crease. Our
creases are also doubly curved because they are toroidal and,
using the Pogorelov viewpoint, we see how they are stretched:
If we return to the strain energy component in Eq. (10) and
reinsert the flexural rigidity D into this expression, we find
that the energy stored in our creases varies as D(r2 /t)1/2 . For
the straight creases in [13], this variation has been shown to
go as D(L/t)1/3 , where L is the axial length of the crease.
In essence, r2 also measures the length of our creases, so
we see a common length-to-thickness ratio but with quite
different power-law exponents. This comparison could be
serendipitous, but it begs an obvious and interesting research
question for future study. One approach might be to combine
what is known about the bending of straight creases with that
of forming conical creases in a generalized way. Finally, we
note that it is not surprising that our single conical crease is
geometrically similar to the repetitive, periodic wrinkles in
buckled thin-walled plates or membranes. Indeed, in [10] the
authors remark from an energy perspective that a characteristic
wavelength requires ingredients equivalent to the bending of a
plate on an elastic foundation: Such problems are well known
to be governed by a fourth-order linear differential equation of
similar form to Eq. (3) [8]. Thus, we would expect the same
nature and dimensional variation in our solution.
ACKNOWLEDGMENTS
The author first worked on this problem during a visiting
research tenure at the University of Pierre and Marie Curie in
Paris. He is extremely grateful to Professor Corrado Maurini
for making this stay and its associated “head-space” happen.
Finally, the author is grateful to one of the anonymous
referees for emphasizing the comparison to problems involving
periodic wrinkling of thin plates and membranes.
[2] E. Zhu, P. Mandal, and C. R. Calladine, Buckling of thin
cylindrical shells: An attempt to resolve a paradox, Int. J. Mech.
Sci. 44, 1583 (2002).
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INVERTED CONES AND THEIR ELASTIC CREASES
PHYSICAL REVIEW E 94, 063002 (2016)
[3] M. Gomez, D. E. Moulton, and D. Vella, The shallow shell
approach to Pogorelov’s problem and the breakdown of ‘mirror
buckling’, Proc. R. Soc. A 453, 20150732 (2016).
[4] S. Knoche and J. Kierfeld, Secondary polygonal instability of buckled spherical shells, Europhys. Lett. 106, 24004
(2014).
[5] P. K. Gupta, A study on inversion of metallic thin-walled conical
shells, Int. J. Crashworthiness 16, 607 (2011).
[6] A. Ramachandran, First Year Ph.D. Report, Department of Materials Science, University of Cambridge, 2013 (unpublished).
[7] ABAQUS, Analysis User’s Manual Version 6.11 (Dassault
Systèmes, Providence, 2011).
[8] C. R. Calladine, Theory of Shell Structures (Cambridge University Press, Cambridge, 1983).
[9] E. H. Mansfield, The Bending and Stretching of Plates
(Cambridge University Press, Cambridge, 1989).
[10] E. Cerda and L. Mahadevan, Geometry and Physics of Wrinkling, Phys. Rev. Lett. 90, 074302 (2003).
[11] Y. W. Wong and S. Pellegrino, Wrinkled membranes. Part II:
Analytical models, J. Mech. Mater. Struct. 1, 27 (2006).
[12] M. Ben Amar and Y. Pomeau, Crumpled paper, Proc. R. Soc. A
453, 729 (1997).
[13] A. E. Lobkovsky and T. A. Witten, Properties of ridges in elastic
membranes, Phys. Rev. E 55, 1577 (1997).
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